Expert: Paul Klarreich - 9/21/2008

Question
This problem I have been struggling with for about the past 30 minutes hopefully you can help.

h(x)=27x^3+6x
    10-27x^3

The 27x^3+6x is divided by 10-27x^3 my browser is not letting me put a line under the numerator.

And I am asked to decompose to f(g(x))=h(x)
and f(x) and g(x)

Thanks for any help you may provide.

Answer
Questioner:   Chris
Category:  Calculus
Private:  No
 
Subject:  College Algebra
Question:  This problem I have been struggling with for about the past 30 minutes hopefully you can help.

h(x)=27x^3+6x
   10-27x^3

The 27x^3+6x is divided by 10-27x^3 my browser is not letting me put a line under the numerator.

And I am asked to decompose to f(g(x))=h(x)
and f(x) and g(x)

Thanks for any help you may provide.
............................................
Hi, Chris,
For your function:

       27x^3 + 6x
h(x) =  ------------
       10 - 27x^3

you want to find some f and g such that  f(g(x)) = h(x)

Obviously, there is a 'trivial' solution, where:

g(x) = h(x) and  f(x) = x,

but that is no fun.  You want to find a sequence of operations that computes h(x), then break that into two 'subsequences'.  Try this:

Since

       3x(9x^2 + 2) << factored
g(x) =  ------------
       10 - 27x^3
 

Take this sequence:  All the stuff up to the final multiplication by 3, then the multiplication by 3:

That is:

       x(9x^2 + 2)  << no 3.
g(x) =  -----------
       10 - 27x^3

and  f(x) = 3x

Try it.  That means

f(g(x)) = 3 g(x) = ... you can take it from here.